Abstract
In the recent years some efforts were made to propose simple and well-behaved fractional derivatives that inherits the classical properties from the first order derivative. Therefore, we propose in this research a new strategy to acquire interval solution of fractional interval differential equations (FIDEs) under interval fractional conformable derivative. This scheme is developed based on a variation of the constant formula to achieve the solution explicitly. The important characteristic of this technique is that it helps us to find a solution with decreasing length of its support which is critical for the solutions based on the interval or fuzzy notions. Two examples are experienced to illustrate our approach and validate it.
Similar content being viewed by others
References
D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional calculus: models and numerical methods (World Scientific, 2012)
I. Podlubny, Fractional differential equations (Academic Press, San Diego, CA, 1999)
O.P. Agrawal, J.A. Tenreiro-Machado, I. Sabatier, Fractional derivatives and their applications, in Nonlinear dynamics (Springer-Verlag, Berlin, 2004), Vol. 38
D. Baleanu, Z.B. Güvenç, J.A. Tenreiro Machado (Eds.) New trends in nanotechnology and fractional calculus applications (Springer, New York, 2010)
I. Stamova, G. Stamov, Functional and impulsive differential equations of fractional order: qualitative analysis and applications (CRC Press, 2017)
I. Stamova, Appl. Math. Comput. 237, 605 (2014)
J.O. Alzabut, T. Abdeljawad, J. Fract. Calc. Appl. 5, 177 (2014)
S. Sun, T. Abdeljawad, J. Alzabut, Discr. Dyn. Nat. Soc. 2013, Article ID 609861 (2013)
X.L. Ding, J.J. Nieto, J. Comput. Nonlinear Dyn. 12, 031018 (2017)
A.B. Malinowska, D.F.M. Torres, Introduction to the fractional calculus of variations (World Scientific Publishing Co Inc., 2012)
Y. Zhou, F. Jiao, Comput. Math. Appl. 59, 1063 (2010)
G. Stamov, I. Stamova, Neural Comput. Appl. 28, 3307 (2016)
G. Stamov, I. Stamova, IMA J. Appl. Math. 80, 1619 (2015)
D. Benson, S.W. Wheatcraft, M.M. Meerschaert, Water Resour. Res. 36, 1413 (2000)
R. Khalil, M. Al Horani, A. Yousef, M. Sababhehb, J. Comput. Appl. Math. 264, 65 (2014)
T. Abdeljawad, J. Comput. Appl. Math. 279, 57 (2015)
S. Markov, Computing 22, 325 (1979)
V. Lupulescu, Fuzzy Sets Syst. 265, 63 (2015)
M.T. Malinowski, Appl. Math. Lett. 24, 2118 (2011)
M.T. Malinowski, Inform. Sci. 213, 94 (2012)
V. Lupulescu, Inform. Sci. 248, 50 (2013)
Y. Chalco-Cano, W.A. Lodwick, B. Bede, Fuzzy Sets Syst. 257, 146 (2014)
L. Stefanini, Fuzzy Sets Syst. 161, 1564 (2010)
S. Salahshour, A. Ahmadian, F. Ismail, D. Baleanu, Optik 130, 273 (2017)
A. Ahmadian, S. Salahshour, C.S. Chan, D. Baleanu, Fuzzy Sets Syst. 331, 47 (2018)
R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, Nonlinear Anal. 72, 2859 (2010)
R.P. Agarwal, S. Arshad, D. O’Regan, V. Lupulescu, Fract. Calc. Appl. Anal. 15, 572 (2012)
S. Arshad, V. Lupulescu, Nonlinear Anal. 74, 85 (2011)
T. Allahviranloo, S. Salahshour, S. Abbasbandy, Soft Comput. 16, 297 (2012)
N.V. Hoa, Fuzzy Sets Syst. 280, 58 (2015)
S. Salahshour, T. Allahviranloo, S. Abbasbandy, D. Baleanu, Adv. Diff. Equ. 2012, 112 (2012)
M.T. Malinowski, Fuzzy Sets Syst. 265, 39 (2015)
S. Salahshour, T. Allahviranloo, S. Abbasbandy, Commun. Nonlinear Sci. Numer. Simul. 17, 1372 (2012)
S. Salahshour, A. Ahmadian, N. Senu, D. Baleanu, P. Agarwal, Entropy 17, 885 (2015)
A. Ahmadian, C.S. Chang, S. Salahshour, IEEE Trans. Fuzzy Syst. 25, 218 (2017)
A. Ahmadian, M. Suleiman, S. Salahshour, D. Baleanu, Adv. Diff. Equ. 2013, 104 (2013)
A. Ahmadian, S. Salahshour, D. Baleanu, H. Amirkhani, R. Yunus, J. Comput. Phys. 294, 562 (2015)
A. Khastan, J. Comput. Appl. Math. 312, 156 (2017)
A. Khastan, Soft Comput. 21, 3503 (2017)
H. Sun, X. Song, Y. Chen, A class of fractional dynamic systems with fuzzy order, in 2010 8th World Congress on Intelligent Control and Automation (WCICA) (IEEE, 2010), pp. 197–201
H. Sun, H. Sheng, Y. Chen, W. Chen, Z. Yu, Chin. Phys. Lett. 30, 046601 (2013)
B. Bede, I.J. Rudas, A.L. Bencsik, Inform. Sci. 177, 1648 (2007)
A. Khastan, Alireza, J.J. Nieto, R. Rodríguez-López, Fuzzy Sets Syst. 177, 20 (2011)
B. Bede, S.G. Gal, Fuzzy Sets Syst. 151, 581 (2005)
M. Chehlabi, T. Allahviranloo, Appl. Soft Comput. 44, 108 (2016)
S. Salahshour, A. Ahmadian, F. Ismail, D. Baleanu, N. Senu, Adv. Mech. Eng. (2015), https://doi.org/10.1177/1687814015619138
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Salahshour, S., Ahmadian, A. & Baleanu, D. Variation of constant formula for the solution of interval differential equations of non-integer order. Eur. Phys. J. Spec. Top. 226, 3501–3512 (2017). https://doi.org/10.1140/epjst/e2018-00064-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2018-00064-2